
theorem Th54:
for I,J be non empty closed_interval Subset of REAL,
 g being PartFunc of [:RNS_Real,RNS_Real:],RNS_Real,
 f be PartFunc of [:REAL,REAL:],REAL
 st [:I,J:] = dom g & g is_continuous_on [:I,J:] & g = f holds
   Integral1(L-Meas,|.R_EAL f.|)|J is PartFunc of REAL,REAL
 & Integral1(L-Meas,R_EAL f)|J is PartFunc of REAL,REAL
proof
    let I,J be non empty closed_interval Subset of REAL,
    g be PartFunc of [:RNS_Real,RNS_Real:],RNS_Real,
    f be PartFunc of [:REAL,REAL:],REAL;
    assume that
A1:  [:I,J:] = dom g and
A2:  g is_continuous_on [:I,J:] and
A3:  g = f;

    set g1 =R_EAL f;

    set T = Integral1(L-Meas,|.g1.|);
A4:dom T = REAL by FUNCT_2:def 1;
    reconsider T0 = T|J as PartFunc of REAL,ExtREAL;

    set RT = Integral1(L-Meas,g1);
A5:dom RT = REAL by FUNCT_2:def 1;
    reconsider RT0 = RT|J as PartFunc of REAL,ExtREAL;

    now let x be object;
     assume x in rng T0; then
     consider y be Element of REAL such that
A6:  y in dom T0 & x = T0.y by PARTFUN1:3;

     reconsider Pg = ProjPMap2(|.g1.|,y) as PartFunc of REAL,REAL
      by Th30;
     (Integral1(L-Meas,|.g1.|)).y = integral(Pg,I)
       by A6,A4,A1,A2,A3,Th49; then
     (Integral1(L-Meas,|.g1.|)).y in REAL by XREAL_0:def 1;
     hence x in REAL by A6,A4,FUNCT_1:49;
    end; then
    rng T0 c= REAL & dom T0 c= REAL;
    hence Integral1(L-Meas,|.R_EAL f.|)|J is PartFunc of REAL,REAL
      by RELSET_1:4;

    now let x be object;
     assume x in rng RT0; then
     consider y be Element of REAL such that
A7: y in dom RT0 & x= RT0.y by PARTFUN1:3;

     reconsider Pg = ProjPMap2(g1,y) as PartFunc of REAL,REAL by Th30;
     (Integral1(L-Meas,R_EAL f)).y = integral(Pg,I)
       by A7,A5,A1,A2,A3,Th43; then
     (Integral1(L-Meas,g1)).y in REAL by XREAL_0:def 1;
     hence x in REAL by A7,A5,FUNCT_1:49;
    end; then
    rng RT0 c= REAL & dom RT0 c= REAL;
    hence Integral1(L-Meas,R_EAL f)|J is PartFunc of REAL,REAL by RELSET_1:4;
end;
